I Can an object accelerate at a constant pace forever?

Considering special relativity, let us take an object accelerating at a rate of 6m/s^2. Let's say that this object is large enough for an observer to securely stand on it while it is accelerating.

Would this object be able to accelerate at this rate forever without reaching the speed of light, and if so, what affects would the observer feel as the velocity reaches the speed of light?

Side question: Let's say a ball is held up 1m above the surface of this object, and is let go. Would the time the ball takes to reach the surface differ if the velocity of the object was at 25% the speed of light compared to 75% the speed of light?

Also, what would happen (if anything), if the object is travelling at a velocity of 299,792,457m/s (one less that the speed of light)?

Side question: Let's say a ball is held up 1m above the surface of this object, and is let go. Would the time the ball takes to reach the surface differ if the velocity of the object was at 25% the speed of light compared to 75% the speed of light?

You may want to come up with some other question that doesn't imply gravity. Otherwise all physical processes take the usual amount of time, measured with the observe's clock.

If, by constant acceleration you mean "as measured in a given inertial frame", the answer is no, it cannot continue indefinitely. If you mean, can a rocket accelerate such that objects inside experience constant g-force forever, the answer is yes (barring the practical constraints mentioned in Vitro's post). In this case, the time for an object to fall from from the top of the rocket to the bottom, as measured by the bottom (there are some subtleties here, which I will gloss over for the moment; suffice it to say, you have to define carefully what yo mean by this because there is time dilation of clocks at the bottom of the rocket compared to the top), will remain the same, always. Specifically, even if the the rocket is moving 1 cm/sec less that c per some initial inertial frame, the top to bottom fall measurement as done in the rocket will be the same as when the rocket was moving at 1 meter/sec per this initial inertial frame.